Numerical solution of nonlinear two-dimensional Volterra integral equation of the second kind in the reproducing kernel space

Abstract

In this article, an effective method is given to solve nonlinear two-dimensional Volterra integral equations of the second kind. First, we find the solution of integral equation in terms of reproducing kernel functions in series, then by truncating the series an approximate solution obtained. In addition, the calculation of Fourier coefficients solution of the integral equation in terms of reproducing kernel functions is notable. Numerical examples are presented, and their results are compared with the analytical solution to demonstrate the validity and applicability of the method.

Approximati of Solutions of 2d Non-Linear Integral Equations in a Binary Hilbert Space

“…Two-dimensional integral equations of the second kind can be used to model a variety of physics and engineering issues. [1][2][3][4][5][6]. We consider a general class of nonlinear Volterra integral equations in two dimensions: u(x ,y)=A(x ,y ,u(x ,y))+ ∬ 𝐹(𝑥, 𝑦, 𝑠, 𝑡, 𝑢(𝑠, 𝑡))𝑑𝑡𝑑𝑠, 𝑥𝑦 𝑎𝑐 (1,1) for all" (x, y) ∈ D = [a, b]×[c, d]"; where "u ∈ ℍ(D)" is an unknown function to be determined, F satisfies the given regularity criterion and A is a given function in reproducing kernel Hilbert space ℍ(D); see Definition (2.6).…”

“…[1][2][3][4][5][6]. We consider a general class of nonlinear Volterra integral equations in two dimensions: u(x ,y)=A(x ,y ,u(x ,y))+ ∬ 𝐹(𝑥, 𝑦, 𝑠, 𝑡, 𝑢(𝑠, 𝑡))𝑑𝑡𝑑𝑠, 𝑥𝑦 𝑎𝑐 (1,1) for all" (x, y) ∈ D = [a, b]×[c, d]"; where "u ∈ ℍ(D)" is an unknown function to be determined, F satisfies the given regularity criterion and A is a given function in reproducing kernel Hilbert space ℍ(D); see Definition (2.6). Recently, reproducing kernel Hilbert space method (RKHSM) has been extensively studied and successfully applied to many real world problems such as in differential and integral equations, fractional differential equations, physics including fractional gas dynamics, advection-diffusion equation, fractional diffusion and advection-dispersion equations [2,[5][6][7][8][9].…”